| Underactuated systems have several irreplaceable advantages in saving energy, reducing costs, reducing cost and improving system reliability compared with fully actuated systems. Underactuated systems are used in fields of aviation, aerospace, marine exploration and etc. As underactuated systems have fewer actuators than degrees of freedoms, control problem of underactuated systems is challenging. Control problem of underactuated systems has been studied intensively by many researchers.Output tracking control problem of a class of underactuated systems is investigated in this paper with nonlinear control methods. The plant for control is ball and plate system. This research is supported by College Doctor Special Scientific Research Fund of China under Grant 20060183006. Name of the project is 'stabilization and trajectory tracking control of the ball and plate system under high speed'. Main topics of this paper are summarized as below.1. Nonlinear control problem is studied for a class of underactuated systems. The underactuated system has one underactuated degree and one fully actuated degree. The underactuated system can not be written into the upper triangular system. If system dynamics to be stabilized is chosen to be state variables as the common backstepping design, it is very difficulty to obtain a stable closed loop system. Concept of contacting function is introduced to solve the problem. Contacting function and its first derivative are selected as the dynamics to be stabilized in the backstepping design. Control Lyapunov functions are constructed step by step, and nonlinear output tracking controller is built recursively.2. As some parameters of the underactuated system may be unknown, uncertain system parameters are added into mathematical model of the underactuated systems above. Nonlinear adaptive control problem of a class of underactuated systems is discussed. And backstepping design procedures are used to stabilize the dynamics of underactuated degree of freedom and fully actuated degree of freedom. Adaptive update laws for the uncertain parameters are also constructed. Uncertain parameters are estimated by overparametrization method.3. As unknown disturbances are inevitable in engineering systems, unstructured uncertainty of the underactuated system is considered. And nonlinear robust output tracking control problem for a class of underactuated systems is investigated. A Priori bounded functions are introduced to reduce the influence of unknown disturbances. Nonlinear robust output tracking controller is constructed recursively for the underactuated system.4. Motion and structure characteristics of the ball and plate system are investigated. Model of the ball and plate system are built including dynamics of ball moving on the plate, stepper motor, transmission sets and friction between the ball and plate. Controllability and observability of the ball and plate system is investigated with geometric theory of nonlinear system. Dynamics discussed in the analysis of controllability and observability is the dynamics of ball moving on the plate.5. Contacts between dynamics of the underactuated systems and motions of the underactuated systems are discussed. Concept of main dynamics is introduced. Relationships between main dynamics and motions of the underactuated systems are studied. And system main dynamics of the ball and plate system is illustrated.6. After mathematical model of the ball and plate system is established, stabilization control problem of the ball and plate system is discussed. The stabilization control problem is investigated on the experimental platform BPVS-JLU-â…¡. Frictions between the ball and the plate are estimated by nonlinear state observer in augmented state space. And then stabilization controller with friction compensation is built with theory of feedback linearization and optimal control. Further, when initial conditions of the ball and plate system vary in a wide range and control inputs are limited by saturation, backstepping design with automatic tuning parameters is proposed for the stabilization control problem. Fuzzy logic is constructed to tune the design parameters automatically in the backstepping control. Parameters of the fuzzy logic are optimized by genetic algorithm under restricted conditions introduced by control saturations.7. Nonlinear output tracking control problem is studied on the experimental platform BPVS-JLU-â…¡. When ball parameters are known, position controller is constructed with variable structure control method first. And then feedback linearization method is used to avoid chattering phenomenon. Adaptive control problem is further investigated for the ball and plate system with structured uncertainty. When radius, mass and inertia of the ball is not specific, nonlinear adaptive controller based on Lyapunov direct method is built first. As the transmission sets of ball and plate system is influenced by backlash nonlinearities, adaptive backstepping design is employed to avoid chattering phenomenon of the nonlinear adaptive controller using Lyapunov direct method. Circle tracking experiments were established on the experimental platform BPVS-JLU-â…¡with expected tracking velocity 6 mm/s, 18 mm/s, 30 mm/s respectively. Five control schemes are used to construct the position controller including backstepping, adaptive backstepping, variable structure control, feedback linearization and fuzzy control. Relationship between expected tracking velocity and control precision of the ball and plate system was discussed for different control schemes. Some experimental phenomenon of the ball and the plate in the tracking control was analyzed. Trajectory tracking experiments established on the ball and plate system experimental platform BPVS-JLU-â…¡have indicated that when nonlinear model of the ball and plate system is considered and position controller is constructed with backstepping or adaptive backstepping design, trajectory tracking control could be done with higher control precision compared with other control schemes.
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